# Why is the accuracy of $\Pr($hypothesis) in Bayes's Theorem less important than apparent?

Source: p 224, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn.
I capitalised miniscules, which the author uses for variables.

I pursue only intuition; please do not answer with formal proofs. Discussing Bayes's Theorem
( $\Pr(H|E)=\frac{\Pr(E|H)\Pr(H)}{\Pr(E)}$ ), the author abbreviates 'evidence' to E and 'hypothesis' to H.

Of course, very often it is difficult or impossible to quantify the "prior" probabilities [ $\Pr(H)$ ] with any accuracy. It is important to realize that this need not matter as much as it might seem. Two factors alleviate the problem. First, even if we assign a range to each figure,

[1.] it may be that all ways of calculating the upshot give a sufficiently similar result. And second, it may be that in the face of enough evidence,

[2.] difference of prior opinion gets swamped. Investigators starting with very different antecedent attitudes to $\Pr(H)$ might end up assigning similarly high values to $\Pr(H \mid E)$, when $E$ becomes impressive enough.

My interpretation of the above as overconfident and presumptuous implies my failure to comprehend it; somehow, I am unpersuaded by 1 and 2. Would these please be explained?

2) basically says "If you receive enough evidence, however unconvinced you were at the start, you must eventually become convinced". Since in real life we actually have a lot of data - medical trials, for instance, get run even when we could in principle know the answers already from extant data - we can afford to be quite imprecise with our priors. (Of course, the assumption here is that we have lots of data. If we are trying to extract the truth from very little data, then it becomes very important to have a reasonable prior, because the updating process has so little effect when the data is so short.)

1 seems to be a rephrasing of 2, to me; 2 seems to be the reason why 1 is true.

The second statement is related to the way that Bayesian probability reacts in the light of new evidence, always converging towards the implied truth.

No matter whether you are skeptical about the relation between cancer and tobacco, or a believer; after seeing a certain amount of reasonable evidence you should change your opinion to fit better the facts.

It is however true that certain specially toxic initial priors can hamper your ability to reason correctly. Perhaps the more simple and radical example of this is when you take as prior $P(A)=0$ or $P(A)=1$.

The problem of selecting reasonable initial prior is difficult , but can be as simple in some special occasions as starting from the total ignorance ($P(A)=1/2$) and letting the data correct our views.

Regarding 1, I do not fully get what he is getting at. Every way of calculating something should give the same result, since it is uniquely determined from the priors. If we have some small discrepancies between our prior and that of our partner's, which is further reduced by evidence updates, then in many cases we can be content with either result, but we have to be careful.

Edit:
You can take A as any statement which is susceptible to be changed by evidence (which is a pretty broad category, though it may exclude logical propositions if we are presupposing logical omniscience). For example, A could be the aforementioned 'Smoking causes cancer'.

• Thanks. Can you please clarify what $A$ means in your 3rd paragraph? – Greek - Area 51 Proposal Feb 7 '16 at 6:29